Class SquaredEuclideanSilhouette


public class SquaredEuclideanSilhouette extends Object
SquaredEuclideanSilhouette computes the average of the Silhouette over all the data of the dataset, which is a measure of how appropriately the data have been clustered.

The Silhouette for each point i is defined as:

$$ s_{i} = \frac{b_{i}-a_{i}}{max\{a_{i},b_{i}\}} $$

which can be rewritten as

$$ s_{i}= \begin{cases} 1-\frac{a_{i}}{b_{i}} & \text{if } a_{i} \leq b_{i} \\ \frac{b_{i}}{a_{i}}-1 & \text{if } a_{i} \gt b_{i} \end{cases} $$

where $a_{i}$ is the average dissimilarity of i with all other data within the same cluster, $b_{i}$ is the lowest average dissimilarity of i to any other cluster, of which i is not a member. $a_{i}$ can be interpreted as how well i is assigned to its cluster (the smaller the value, the better the assignment), while $b_{i}$ is a measure of how well i has not been assigned to its "neighboring cluster", i.e. the nearest cluster to i.

Unfortunately, the naive implementation of the algorithm requires to compute the distance of each couple of points in the dataset. Since the computation of the distance measure takes D operations - if D is the number of dimensions of each point, the computational complexity of the algorithm is O(N^2^*D), where N is the cardinality of the dataset. Of course this is not scalable in N, which is the critical number in a Big Data context.

The algorithm which is implemented in this object, instead, is an efficient and parallel implementation of the Silhouette using the squared Euclidean distance measure.

With this assumption, the total distance of the point X to the points $C_{i}$ belonging to the cluster $\Gamma$ is:

$$ \sum\limits_{i=1}^N d(X, C_{i} ) = \sum\limits_{i=1}^N \Big( \sum\limits_{j=1}^D (x_{j}-c_{ij})^2 \Big) = \sum\limits_{i=1}^N \Big( \sum\limits_{j=1}^D x_{j}^2 + \sum\limits_{j=1}^D c_{ij}^2 -2\sum\limits_{j=1}^D x_{j}c_{ij} \Big) = \sum\limits_{i=1}^N \sum\limits_{j=1}^D x_{j}^2 + \sum\limits_{i=1}^N \sum\limits_{j=1}^D c_{ij}^2 -2 \sum\limits_{i=1}^N \sum\limits_{j=1}^D x_{j}c_{ij} $$

where $x_{j}$ is the j-th dimension of the point X and $c_{ij}$ is the j-th dimension of the i-th point in cluster $\Gamma$.

Then, the first term of the equation can be rewritten as:

$$ \sum\limits_{i=1}^N \sum\limits_{j=1}^D x_{j}^2 = N \xi_{X} \text{ , with } \xi_{X} = \sum\limits_{j=1}^D x_{j}^2 $$

where $\xi_{X}$ is fixed for each point and it can be precomputed.

Moreover, the second term is fixed for each cluster too, thus we can name it $\Psi_{\Gamma}$

$$ \sum\limits_{i=1}^N \sum\limits_{j=1}^D c_{ij}^2 = \sum\limits_{i=1}^N \xi_{C_{i}} = \Psi_{\Gamma} $$

Last, the third element becomes

$$ \sum\limits_{i=1}^N \sum\limits_{j=1}^D x_{j}c_{ij} = \sum\limits_{j=1}^D \Big(\sum\limits_{i=1}^N c_{ij} \Big) x_{j} $$

thus defining the vector

$$ Y_{\Gamma}:Y_{\Gamma j} = \sum\limits_{i=1}^N c_{ij} , j=0, ..., D $$

which is fixed for each cluster $\Gamma$, we have

$$ \sum\limits_{j=1}^D \Big(\sum\limits_{i=1}^N c_{ij} \Big) x_{j} = \sum\limits_{j=1}^D Y_{\Gamma j} x_{j} $$

In this way, the previous equation becomes

$$ N\xi_{X} + \Psi_{\Gamma} - 2 \sum\limits_{j=1}^D Y_{\Gamma j} x_{j} $$

and the average distance of a point to a cluster can be computed as

$$ \frac{\sum\limits_{i=1}^N d(X, C_{i} )}{N} = \frac{N\xi_{X} + \Psi_{\Gamma} - 2 \sum\limits_{j=1}^D Y_{\Gamma j} x_{j}}{N} = \xi_{X} + \frac{\Psi_{\Gamma} }{N} - 2 \frac{\sum\limits_{j=1}^D Y_{\Gamma j} x_{j}}{N} $$

Thus, it is enough to precompute: the constant $\xi_{X}$ for each point X; the constants $\Psi_{\Gamma}$, N and the vector $Y_{\Gamma}$ for each cluster $\Gamma$.

In the implementation, the precomputed values for the clusters are distributed among the worker nodes via broadcasted variables, because we can assume that the clusters are limited in number and anyway they are much fewer than the points.

The main strengths of this algorithm are the low computational complexity and the intrinsic parallelism. The precomputed information for each point and for each cluster can be computed with a computational complexity which is O(N/W), where N is the number of points in the dataset and W is the number of worker nodes. After that, every point can be analyzed independently of the others.

For every point we need to compute the average distance to all the clusters. Since the formula above requires O(D) operations, this phase has a computational complexity which is O(C*D*N/W) where C is the number of clusters (which we assume quite low), D is the number of dimensions, N is the number of points in the dataset and W is the number of worker nodes.

  • Constructor Details

    • SquaredEuclideanSilhouette

      public SquaredEuclideanSilhouette()
  • Method Details

    • registerKryoClasses

      public static void registerKryoClasses(SparkContext sc)
      This method registers the class SquaredEuclideanSilhouette.ClusterStats for kryo serialization.

      sc - SparkContext to be used
    • computeClusterStats

      public static scala.collection.immutable.Map<Object,SquaredEuclideanSilhouette.ClusterStats> computeClusterStats(Dataset<Row> df, String predictionCol, String featuresCol, String weightCol)
      The method takes the input dataset and computes the aggregated values about a cluster which are needed by the algorithm.

      df - The DataFrame which contains the input data
      predictionCol - The name of the column which contains the predicted cluster id for the point.
      featuresCol - The name of the column which contains the feature vector of the point.
      weightCol - The name of the column which contains the instance weight.
      A Map which associates each cluster id to a SquaredEuclideanSilhouette.ClusterStats object (which contains the precomputed values N, $\Psi_{\Gamma}$ and $Y_{\Gamma}$ for a cluster).
    • computeSilhouetteCoefficient

      public static double computeSilhouetteCoefficient(Broadcast<scala.collection.immutable.Map<Object,SquaredEuclideanSilhouette.ClusterStats>> broadcastedClustersMap, Vector point, double clusterId, double weight, double squaredNorm)
      It computes the Silhouette coefficient for a point.

      broadcastedClustersMap - A map of the precomputed values for each cluster.
      point - The Vector representing the current point.
      clusterId - The id of the cluster the current point belongs to.
      weight - The instance weight of the current point.
      squaredNorm - The $\Xi_{X}$ (which is the squared norm) precomputed for the point.
      The Silhouette for the point.
    • computeSilhouetteScore

      public static double computeSilhouetteScore(Dataset<?> dataset, String predictionCol, String featuresCol, String weightCol)
      Compute the Silhouette score of the dataset using squared Euclidean distance measure.

      dataset - The input dataset (previously clustered) on which compute the Silhouette.
      predictionCol - The name of the column which contains the predicted cluster id for the point.
      featuresCol - The name of the column which contains the feature vector of the point.
      weightCol - The name of the column which contains instance weight.
      The average of the Silhouette values of the clustered data.
    • pointSilhouetteCoefficient

      public static double pointSilhouetteCoefficient(scala.collection.immutable.Set<Object> clusterIds, double pointClusterId, double weightSum, double weight, scala.Function1<Object,Object> averageDistanceToCluster)
    • overallScore

      public static double overallScore(Dataset<Row> df, Column scoreColumn, Column weightColumn)